The Value and Its Origin

GANITA, Vol. 65, 2016, 77-86 77 PI -THE VALUE AND ITS ORIGIN Dr.Debangana Rajput, Associate Professor Department of Mathematics, Sri JNPG College University of Lucknow, Lucknow email: [email protected] Abstract Due to the importance of π in science, particularly in physics and mathematics, a number of researchers have contributed to compute its value in a reasonable precision. This article highlights the progress made by researchers and mathematicians in calculating a better approximation of its value using different methods and tools. A historical perspective has also been discussed here. Subject class [2010]:11-XX Keywords: pi, irrational number, transcendental number. 1 INTRODUCTION PI(Π)is the sixteenth letter of the Greek alphabet. It is used to represent a special mathematical constant defined as the ratio of the circumference of a circle to its diameter, which is approximately equal to 3.1415926535897932 (upto 16 decimal places). π is an irrational and transcendental number, which means it cannot be expressed as a ratio of two integers and has an infinite number of digits in its digital representation. Further, it is not the solution of any non-integer polynomial with rational coefficients. The computation of π is one of the ancient topics of mathematics that is still of serious interest to modern mathematical research. Due to its importance, a special day called 'Pi day' is also celebrated on March 14th because of its date, i.e. 3/14, which contains first three significant digits in approximation of π. ’Π Approximation Day' is also observed on July 22nd i.e.22/7 format because it is the common approximation of π. 2 HISTORICAL PERSPECTIVE The earliest recorded calculations of π date back to the babylonian,Egyptian and Hebrew civilizations. The ratio of the circumference to the diameter equal to 3 even appears in a verse in the Hebrew Bible(written around the 4th century BC) proving that 3 was considered as an approximation for what we know today as π. A verse of the Hebrew Bible reads [1]: 'And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about.' (They describe a ceremonial pool in the Temple of Solomon that had a diameter of ten cubits and a circumference of thirty cubits, and the verses imply that the pool is circular if π is about three.)A clay tablet from the Babylonian[2] civilization dated 1900-1600 BC and unearthed in 1936 at Susa, 200 miles away from Babylon, implicates a value of π as 78 Rajput: PI -THE VALUE AND ITS ORIGIN 1 3 8 =3.125.3.1250. In Egypt, the Rhind Papyrus , a work on mathematics dating back to 1650 BC, finds the area of a square having eight-ninths of the circle's diameter as a side. It can be shown that this empirical formula is equivalent to taking π = 3.1604[2]. It was obtained by squaring the circle. Greek scientist Archimedes of Syracuse[3;21] (290-211BC), developed a polygon approach to approximate the value af π and found the upper and 10 1 lower bounds of π as 3 71 < π < 3 7 i.e 3.1408 < π < 3.1429. His method was devised around 250 BC and dominated for over 800 years. Around 150 AD,Greek-Roman scientist Ptolemy[8;22] (90-168AD) gave a value of 3.1416 in his Almagest using Archimedes's limits. A Chinese mathematician Zu Chongzhi[5].(429-500AD) concluded that the value of pi falls between 3.1415926 and 3.1415927; and therefore he became the first scientist in the world who calculated the value of π to seven decimal places. Zu Chongzhi approximated the value of pi as 355/113, which was called milv(close ratio). The value was the most accurate in the world at that time, and Japanese scientists respectfully called it the "Zu Chongzhi [5] 355 Ratio". TsuCh'ung-chih (430-501AD) and his son computed π = 133 approximately equal to 3.1415929. The Hindu mathematician Aryabhata[9] in his work Ganitapada[7],approximated value of pi as 62,832/20,000 = 3.1416 (as opposed to Archimehes' 'inaccurate' 22/7 which was frequently used), but he apparently never used it,nor did anyone else for several centuries. Another Hindu Mathematician Brahamgupt[8] put forward the concept that the value of π approaches to square root of 10. By the 9th century, mathematics and science prospered in Arab cultures. The Arabian mathematician, Mohammed bin Musa al'Khwarizmi[10] born around 800 BC, and known as 'father of algebra' attempted to calaulate π approximately equal to 31/ 7. During the 16th and 17th centuries the calculation of π was revolutionized by the development of techniques of infinite series especially by the mathematicians from Europe. They calculated π with greater precision. The French mathematician Francmit Viete[11](1540- 1603) gave his final result as 3.1415926535 < π < 3.1415926537 . Viétebecame the first man in history to describe π using an infinite product. In 1593, Adrianus Romanus[6](1561- 1615) computed π to 17 digits after the decimal, of which 15 were correct. Just three years later, a German named Ludolph Van Ceulen [16](1540-1610) presented 35 digits using a polygon of 262 sides. Van Ceulen spent a great part of his life working on π. When he died in 1610 he had accurately found 35 digits. Up to this time, there was no symbol to denote the ratio of a circle's circumference to [16] π its diameter. In 1647 William Oughtred (1575-1660) used δ to represent the circumference of a given circle, so that his π varied according to the circle's diameter, rather than considering as constant as it is known today. In those days the circumference of a circle was known as `periphery', so Greek equivalent for `p' that is `π' was used to denote it. In 1706, William Jones[2;12](1675-1749), a Welsh mathematician used the symbol π for the first time in his book `Synopsis Palmariorum Matheseos'. He mentioned that the ratio of a circle's circumference to its diameter can never be expressed in numbers, though he did not prove it. Upto this time irrationality of π was not known. This symbol of π was not accepted immediately but in 1737 when Leonhard Euler[8] began using the symbol π for pi, then it was quickly accepted. In 1650, John Wallis[13](1616-1703) derived a formula for 4/π which is simplified to GANITA, Vol. 65, 2016, 77-86 79 π/2 using infinite series. Around 1682, Gottfried Leibnitz[14](1646-1716) pointed out that since tan π/4 = 1,it was very simple to calculate π, but to get only two decimal places 300 terms of the series were required and 10,000 terms were required for 4 decimal places. To compute 100 digits, one has to calculate more terms than there are particles in the universe. In 1706, John Machin[14](1680-1751), a professor of astronomy in London, computed 100 places by hand using his new formula. During this time the aim to calculate the value of π was to see whether the digits of π repeat or not to declare it as the ratio of two integers. But around 1768, Johann H. Lambert[4] (1728-1777) proved that π is irrational but could not satisfy everyone. In 1794, Adrien Marie Legendre[15] (1752-1833)gave a rigorous proof for the same. In 1882, Ferdinand von Lindemann[14](1852-1939) proved the transcendence of π. That means it can never be a solution to any finite polynomial with whole number coefficients, which was the answer to the most intriguing question of Greeks, whether a circle could be squared with the help of ruler or compass. In 1873, an Englishman named William Shanks[14](1812-1882) calculated 707 places of π in fifteen years. In 1945 Daniel F. Ferguson[20](1841-1916) discovered the error in Shank's calculation at 528th place onwards. After two years in 1947, he raised the total to 808(accurate) decimal digits. This was the last best approximation using pen and paper. One and a half years later, Levi Smith and John Wrench[20](1911-2009) hit the 1000-digit- mark,using a gear driven calculator. With the development of computers, mathematician made efforts to calculate more and more precise value of π. In 1949, another breakthrough emerged, but it was not mathematical in nature; it was the speed with which the calculations could be done. The ENIAC (Electronic Numerical Integrator and Computer) was finally completed and made functional. So John von Neumann and Reitwisener[16]together with their team used ENIAC to calculate pi. The machine calculated 2037 digits in just seventy hours. John Wrench and Daniel Shanks[20] found 100,265 digits in 1961 using IBM 7090, and the one-million-mark was surpassed in 1973. Gregory Chudnovsky[26] brothers broke the one-billion-barrier in August 1989. In 1997, Kanada and Takahashi[4] calculated 51.5 billion. In 1999 by Kanada and Takahashi calculated 68,719,470,000 digits. In 2010, 5 trillion digits were calculated by Yee and Kondo[27]. The current record for calculating π, as of 2013 is to 12.1 trillion digits by Kondo and Yee in decimal system. The fraction 22/7 which is generally used as an approximation of π is accurate to 0.04025%. The digits of π have no specific pattern, and are statistically random in occurrence.

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